1
00:00:00,000 --> 00:00:00,000

2
00:00:00,000 --> 00:00:07,740
PROFESSOR: Hi everybody.

3
00:00:07,740 --> 00:00:09,280
Welcome back to recitation.

4
00:00:09,280 --> 00:00:12,100
In lecture you did a bunch of
examples of related rates

5
00:00:12,100 --> 00:00:16,550
problems. So I have a couple
more for you to do today.

6
00:00:16,550 --> 00:00:21,490
So here we've got OK, so we've
got air being blown into a

7
00:00:21,490 --> 00:00:24,950
spherical balloon at a
rate of 1,000 cubic

8
00:00:24,950 --> 00:00:26,690
centimeters per second.

9
00:00:26,690 --> 00:00:30,780
So the question is, how fast is
the radius growing when the

10
00:00:30,780 --> 00:00:32,930
radius is equal to
8 centimeters?

11
00:00:32,930 --> 00:00:35,400
And then OK, so I've got a
second question, which is, how

12
00:00:35,400 --> 00:00:38,810
fast is the surface area growing
at that same time?

13
00:00:38,810 --> 00:00:41,810
So why don't you take a few
minutes, work this one out for

14
00:00:41,810 --> 00:00:44,140
yourself, come back, and we'll
work it out together.

15
00:00:44,140 --> 00:00:51,590

16
00:00:51,590 --> 00:00:52,020
All right.

17
00:00:52,020 --> 00:00:52,830
Welcome back.

18
00:00:52,830 --> 00:00:57,770
So this question, like all
related rates questions, has

19
00:00:57,770 --> 00:00:59,890
the property that the calculus
is typically very

20
00:00:59,890 --> 00:01:04,120
straightforward, but that
there's some geometric or

21
00:01:04,120 --> 00:01:05,040
algebraic setup.

22
00:01:05,040 --> 00:01:08,410
So in this case, it's straight
up geometry.

23
00:01:08,410 --> 00:01:13,890
So we have a balloon, we know
it's a perfect sphere, we know

24
00:01:13,890 --> 00:01:16,540
how fast the volume
is changing.

25
00:01:16,540 --> 00:01:19,050
So OK, but we need to know how
fast the radius is changing.

26
00:01:19,050 --> 00:01:22,100
So in order to do that we need
to figure out a relationship

27
00:01:22,100 --> 00:01:23,910
between the radius
and the volume.

28
00:01:23,910 --> 00:01:26,530
And then we can just do implicit
differentiation like

29
00:01:26,530 --> 00:01:27,540
we've been doing.

30
00:01:27,540 --> 00:01:31,580
So for example, OK,
so for a sphere--

31
00:01:31,580 --> 00:01:34,020
the setup is not so bad
for this first one--

32
00:01:34,020 --> 00:01:41,970
so we know that for a sphere the
volume is equal to 4/3 pi

33
00:01:41,970 --> 00:01:43,670
times the radius cubed.

34
00:01:43,670 --> 00:01:47,620
So that's the fundamental
relationship between the

35
00:01:47,620 --> 00:01:50,410
volume and radius of a sphere,
and it's true for every sphere

36
00:01:50,410 --> 00:01:52,920
everywhere in Euclidean space.

37
00:01:52,920 --> 00:01:58,060
And we're given also, that the
volume is changing at a

38
00:01:58,060 --> 00:02:03,620
constant rate of 1,000
centimeters cubed per second.

39
00:02:03,620 --> 00:02:07,470
So dv dt is just given
to be 1,000.

40
00:02:07,470 --> 00:02:10,650
You know, leave off the
units at this point.

41
00:02:10,650 --> 00:02:13,610
So the question is,
what is dr dt?

42
00:02:13,610 --> 00:02:14,760
That's what we're trying
to figure out.

43
00:02:14,760 --> 00:02:17,840
We're trying to figure out how
fast the radius is changing at

44
00:02:17,840 --> 00:02:21,260
the moment when the radius
is equal to 8 centimers.

45
00:02:21,260 --> 00:02:22,600
So how can we do that?

46
00:02:22,600 --> 00:02:24,530
Well, this fundamental

47
00:02:24,530 --> 00:02:25,790
relationship, it's an identity.

48
00:02:25,790 --> 00:02:26,850
It always holds.

49
00:02:26,850 --> 00:02:30,020
So that means we can
differentiate it.

50
00:02:30,020 --> 00:02:34,180
So if we take the derivative of
this identity, well, V on

51
00:02:34,180 --> 00:02:40,110
the left just becomes dv dt.

52
00:02:40,110 --> 00:02:43,880
And on the right we want to do
implicit differentation.

53
00:02:43,880 --> 00:02:46,930
So here r is changing with
respect to time.

54
00:02:46,930 --> 00:02:48,330
r is a function of t.

55
00:02:48,330 --> 00:02:48,480
Well, OK.

56
00:02:48,480 --> 00:02:50,500
So 4/3 pi is a constant.

57
00:02:50,500 --> 00:02:53,520
So that when we differentiate
nothing happens.

58
00:02:53,520 --> 00:02:54,560
4/3 pi.

59
00:02:54,560 --> 00:02:58,380
And so now we differentiate r
cubed with respect to t, so

60
00:02:58,380 --> 00:03:07,160
that gives us 3r squared
times dr dt.

61
00:03:07,160 --> 00:03:09,790
So that's just the chain
rule in action there.

62
00:03:09,790 --> 00:03:12,955
And now what we want
is this dr dt.

63
00:03:12,955 --> 00:03:14,550
Right?

64
00:03:14,550 --> 00:03:17,830
That's the thing that we're
looking for, is how fast the

65
00:03:17,830 --> 00:03:18,760
radius is growing.

66
00:03:18,760 --> 00:03:20,310
So that's dr dt.

67
00:03:20,310 --> 00:03:23,800
And we want it at the moment
when r is equal to 8.

68
00:03:23,800 --> 00:03:33,010
So when r is equal to 8 this
implies that-- well, OK, so dv

69
00:03:33,010 --> 00:03:38,220
dt is 1,000 always--

70
00:03:38,220 --> 00:03:39,450
and it implies that--

71
00:03:39,450 --> 00:03:46,270
OK, so it's equal to 1,000 is
equal to 4/3 pi times 3 times

72
00:03:46,270 --> 00:03:53,410
8 squared times dr dt.

73
00:03:53,410 --> 00:03:59,070
So at this moment that we're
interested in, we have this

74
00:03:59,070 --> 00:04:02,050
equation to solve for dr dt,
and this is a nice, simple

75
00:04:02,050 --> 00:04:02,900
equation to solve.

76
00:04:02,900 --> 00:04:05,310
You just divide through by
everything on the right hand

77
00:04:05,310 --> 00:04:06,990
side other than dr dt.

78
00:04:06,990 --> 00:04:13,600
So this implies that dr dt is
equal to-- well, OK, so I have

79
00:04:13,600 --> 00:04:16,000
to divide 1,000 by
all this stuff.

80
00:04:16,000 --> 00:04:21,096
I think it works out to
something like 125 over 32 pi.

81
00:04:21,096 --> 00:04:23,360
All right.

82
00:04:23,360 --> 00:04:25,010
So that's the exact value.

83
00:04:25,010 --> 00:04:27,160
Maybe you're interested in
sort of knowing about how

84
00:04:27,160 --> 00:04:28,670
large this is.

85
00:04:28,670 --> 00:04:31,470
So 32 pi is pretty close
to 100, so this

86
00:04:31,470 --> 00:04:35,750
is about 1.2 something.

87
00:04:35,750 --> 00:04:36,270
So, all right.

88
00:04:36,270 --> 00:04:36,900
So there we go.

89
00:04:36,900 --> 00:04:38,530
So that answers the
first question.

90
00:04:38,530 --> 00:04:44,160
At that moment the radius is
growing at a rate of 125 over

91
00:04:44,160 --> 00:04:47,010
32 pi centimeters per second.

92
00:04:47,010 --> 00:04:47,320
OK.

93
00:04:47,320 --> 00:04:49,240
So now how about the second
question that we've got here?

94
00:04:49,240 --> 00:04:50,610
What about the surface area?

95
00:04:50,610 --> 00:04:54,965
So again, we know how fast now
the radius is changing and we

96
00:04:54,965 --> 00:04:56,540
know how fast the volume
is changing.

97
00:04:56,540 --> 00:04:58,900
So in order to figure out how
fast the surface area is

98
00:04:58,900 --> 00:05:01,980
changing, we need something that
relates the surface area

99
00:05:01,980 --> 00:05:04,300
to either the volume
or the radius.

100
00:05:04,300 --> 00:05:07,320
Now the relationship between
surface area and volume is

101
00:05:07,320 --> 00:05:09,960
something that we could sort of
work out if we had to, but

102
00:05:09,960 --> 00:05:12,410
it's a lot easier to write down
the surface area in terms

103
00:05:12,410 --> 00:05:12,910
of the radius.

104
00:05:12,910 --> 00:05:14,160
So let's do that.

105
00:05:14,160 --> 00:05:16,920

106
00:05:16,920 --> 00:05:19,660
So we have, I'm going to use the
letter S to denote surface

107
00:05:19,660 --> 00:05:20,930
area of a sphere.

108
00:05:20,930 --> 00:05:23,640
So again, it's a general
identity, you know, a

109
00:05:23,640 --> 00:05:27,480
geometric fact that the surface
area of a sphere is

110
00:05:27,480 --> 00:05:31,960
equal to 4 pi times the
radius squared.

111
00:05:31,960 --> 00:05:34,000
And this is always true.

112
00:05:34,000 --> 00:05:37,530
And now the thing that we want
is the rate of change of the

113
00:05:37,530 --> 00:05:38,400
surface area.

114
00:05:38,400 --> 00:05:40,850
So the rate of change
is the derivative.

115
00:05:40,850 --> 00:05:45,570
So we want to compute the
derivative here, ds dt.

116
00:05:45,570 --> 00:05:46,580
So OK, so we just do it.

117
00:05:46,580 --> 00:05:52,500
So ds dt is equal to- well,
4 pi hangs around.

118
00:05:52,500 --> 00:05:55,180
And again, we differentiate
r squared.

119
00:05:55,180 --> 00:05:57,080
r is a function of t,
so we have to use

120
00:05:57,080 --> 00:05:58,520
the chain rule here.

121
00:05:58,520 --> 00:06:06,270
So this is times
2r times dr dt.

122
00:06:06,270 --> 00:06:07,890
So this is an identity.

123
00:06:07,890 --> 00:06:09,510
So this is true always.

124
00:06:09,510 --> 00:06:12,660
And now we want to know, at
this particular moment in

125
00:06:12,660 --> 00:06:16,055
time, when r is equal
to 8, what is ds dt?

126
00:06:16,055 --> 00:06:18,920
And in order figure that out,
well OK, we just have to be

127
00:06:18,920 --> 00:06:22,350
able to plug in for
everything else.

128
00:06:22,350 --> 00:06:28,070
So when r equals 8-- all right,
well luckily, now if we

129
00:06:28,070 --> 00:06:30,430
were just starting this problem
from scratch here,

130
00:06:30,430 --> 00:06:31,250
we'd have a problem.

131
00:06:31,250 --> 00:06:33,810
Which is we wouldn't know
what dr dt was.

132
00:06:33,810 --> 00:06:36,800
But luckily, we've already
figured it out right, so in

133
00:06:36,800 --> 00:06:37,830
the first part of the problem.

134
00:06:37,830 --> 00:06:45,030
So we know that when r is equal
to 8, dr dt is equal to

135
00:06:45,030 --> 00:06:49,550
125 over 32 pi.

136
00:06:49,550 --> 00:06:50,750
Did I copy that right?

137
00:06:50,750 --> 00:06:51,490
Yes, I did.

138
00:06:51,490 --> 00:06:52,740
OK.

139
00:06:52,740 --> 00:06:55,610

140
00:06:55,610 --> 00:06:58,920
So OK, so in this case, the
equation we have to solve is

141
00:06:58,920 --> 00:07:00,180
just completely straightforward.

142
00:07:00,180 --> 00:07:03,400
We just plug in the values and
it's already solved for us.

143
00:07:03,400 --> 00:07:04,760
So that's nice.

144
00:07:04,760 --> 00:07:11,970
So that we get that ds dt at
that moment is equal to--

145
00:07:11,970 --> 00:07:21,270
well, it's 4 pi times 2
times 8 is the radius

146
00:07:21,270 --> 00:07:28,610
times 125 over 32 pi.

147
00:07:28,610 --> 00:07:29,625
Oh boy, and all right.

148
00:07:29,625 --> 00:07:32,080
So we can work this out
if we want, I guess.

149
00:07:32,080 --> 00:07:36,980
That's 32 pi to cancel, so
that's equal to 250.

150
00:07:36,980 --> 00:07:40,070
And I guess the units there,
better be centimeters squared

151
00:07:40,070 --> 00:07:41,060
per second.

152
00:07:41,060 --> 00:07:45,100
So at this moment in time the
surface area is growing by 250

153
00:07:45,100 --> 00:07:47,130
centimeters squared
per second.

154
00:07:47,130 --> 00:07:50,140
So that's all we had to do.

155
00:07:50,140 --> 00:07:51,860
So we're all set.